Function of several real variables

In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The “input” variables take real values, while the “output”, also called the “value of the function”, may be real or complex. However, the study of the complex-valued functions may be easily reduced to the study of the real-valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real-valued functions will be considered in this article.

The domain of a function of n variables is the subset of {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n} for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonempty open subset of {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n}.

1 General definition

n = 1

n = 2

n = 3

Functions f(x1, x2, …, xn) of n variables, plotted as graphs in the space Rn + 1. The domains are the red n-dimensional regions, the images are the purple n-dimensional curves.

A real-valued function of n real variables is a function that takes as input n real numbers, commonly represented by the variables x1, x2, …, xn, for producing another real number, the value of the function, commonly denoted f(x1, x2, …, xn). For simplicity, in this article a real-valued function of several real variables will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.

Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable are taken in a subset X of Rn, the domain of the function, which is always supposed to contain an open subset of Rn. In other words, a real-valued function of n real variables is a function

{\displaystyle f:X\to \mathbb {R} }{\displaystyle f:X\to \mathbb {R} }
such that its domain X is a subset of Rn that contains a nonempty open set.

An element of X being an n-tuple (x1, x2, …, xn) (usually delimited by parentheses), the general notation for denoting functions would be f((x1, x2, …, xn)). The common usage, much older than the general definition of functions between sets, is to not use double parentheses and to simply write f(x1, x2, …, xn).

It is also common to abbreviate the n-tuple (x1, x2, …, xn) by using a notation similar to that for vectors, like boldface x, underline x, or overarrow x→. This article will use bold.

A simple example of a function in two variables could be:

{\displaystyle {\begin{aligned}&V:X\to \mathbb {R} \&X=\left{(A,h)\in \mathbb {R} ^{2}\mid A>0,h>0\right}\&V(A,h)={\frac {1}{3}}Ah\end{aligned}

}}{\displaystyle {\begin{aligned}&V:X\to \mathbb {R} \&X=\left{(A,h)\in \mathbb {R} ^{2}\mid A>0,h>0\right}\&V(A,h)={\frac {1}{3}}Ah\end{aligned}}}
which is the volume V of a cone with base area A and height h measured perpendicularly from the base. The domain restricts all variables to be positive since lengths and areas must be positive.

For an example of a function in two variables:

{\displaystyle {z:R2→R&z(x,y)=ax+by

}}{\displaystyle {z:R2→R&z(x,y)=ax+by}}
where a and b are real non-zero constants. Using the three-dimensional Cartesian coordinate system, where the xy plane is the domain R2 and the z axis is the codomain R, one can visualize the image to be a two-dimensional plane, with a slope of a in the positive x direction and a slope of b in the positive y direction. The function is well-defined at all points (x, y) in R2. The previous example can be extended easily to higher dimensions:

{\displaystyle {z:Rp→R&z(x1,x2,…,xp)=a1x1+a2x2+⋯+apxp

}}{\displaystyle {z:Rp→R&z(x1,x2,…,xp)=a1x1+a2x2+⋯+apxp}}
for p non-zero real constants a1, a2, …, ap, which describes a p-dimensional hyperplane.

The Euclidean norm:

{\displaystyle f({\boldsymbol {x}})=|{\boldsymbol {x}}|={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}}f(\boldsymbol{x})=|\boldsymbol{x}| = \sqrt{x_1^2 + \cdots + x_n^2}
is also a function of n variables which is everywhere defined, while

{\displaystyle g({\boldsymbol {x}})={\frac {1}{f({\boldsymbol {x}})}}}g(\boldsymbol{x})=\frac{1}{f(\boldsymbol{x})}
is defined only for x ≠ (0, 0, …, 0).

For a non-linear example function in two variables:

{\displaystyle {\begin{aligned}&z:X\to \mathbb {R} \&X=\left{(x,y)\in \mathbb {R} ^{2},:,x{2}+y^{2}\leq 8,x\neq 0,y\neq 0\right}\&z(x,y)={\frac {1}{2xy}}{\sqrt {x^{2}+y{2}}}\end{aligned}}}{\displaystyle {\begin{aligned}&z:X\to \mathbb {R} \&X=\left{(x,y)\in \mathbb {R} ^{2},:,x{2}+y^{2}\leq 8,x\neq 0,y\neq 0\right}\&z(x,y)={\frac {1}{2xy}}{\sqrt {x^{2}+y{2}}}\end{aligned}}}
which takes in all points in X, a disk of radius √8 “punctured” at the origin (x, y) = (0, 0) in the plane R2, and returns a point in R. The function does not include the origin (x, y) = (0, 0), if it did then f would be ill-defined at that point. Using a 3d Cartesian coordinate system with the xy-plane as the domain R2, and the z axis the codomain R, the image can be visualized as a curved surface.

The function can be evaluated at the point (x, y) = (2, √3) in X:

{\displaystyle z\left(2,{\sqrt {3}}\right)={\frac {1}{2\cdot 2\cdot {\sqrt {3}}}}{\sqrt {\left(2\right)^{2}+\left({\sqrt {3}}\right)^{2}}}={\frac {1}{4{\sqrt {3}}}}{\sqrt {7}},}{\displaystyle z\left(2,{\sqrt {3}}\right)={\frac {1}{2\cdot 2\cdot {\sqrt {3}}}}{\sqrt {\left(2\right)^{2}+\left({\sqrt {3}}\right)^{2}}}={\frac {1}{4{\sqrt {3}}}}{\sqrt {7}},}
However, the function couldn’t be evaluated at, say

{\displaystyle (x,y)=(65,{\sqrt {10}}),\Rightarrow ,x^{2}+y{2}=(65)^{2}+({\sqrt {10}})^{2}>8}(x,y) = (65,\sqrt{10}) , \Rightarrow , x^2 + y^2 = (65)^2 + (\sqrt{10})^2 > 8
since these values of x and y do not satisfy the domain’s rule.

1.1 Image

The image of a function f(x1, x2, …, xn) is the set of all values of f when the n-tuple (x1, x2, …, xn) runs in the whole domain of f. For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.

The preimage of a given real number c is called a level set. It is the set of the solutions of the equation f(x1, x2, …, xn) = c.

1.2 Domain

The domain of a function of several real variables is a subset of Rn that is sometimes, but not always, explicitly defined. In fact, if one restricts the domain X of a function f to a subset Y ⊂ X, one gets formally a different function, the restriction of f to Y, which is denoted {\displaystyle f|{Y}}{\displaystyle f|{Y}}. In practice, it is often (but not always) not harmful to identify f and {\displaystyle f|{Y}}{\displaystyle f|{Y}}, and to omit the restrictor |Y.

Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity or by analytic continuation.

Moreover, many functions are defined in such a way that it is difficult to specify explicitly their domain. For example, given a function f, it may be difficult to specify the domain of the function {\displaystyle g({\boldsymbol {x}})=1/f({\boldsymbol {x}}).}{\displaystyle g({\boldsymbol {x}})=1/f({\boldsymbol {x}}).} If f is a multivariate polynomial, (which has {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n} as a domain), it is even difficult to test whether the domain of g is also {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n}. This is equivalent to test whether a polynomial is always positive, and is the object of an active research area (see Positive polynomial).

1.3 Algebraic structure

The usual operations of arithmetic on the reals may be extended to real-valued functions of several real variables in the following way:

For every real number r, the constant function
{\displaystyle (x_{1},\ldots ,x_{n})\mapsto r}{\displaystyle (x_{1},\ldots ,x_{n})\mapsto r}
is everywhere defined.
For every real number r and every function f, the function:
{\displaystyle rf:(x_{1},\ldots ,x_{n})\mapsto rf(x_{1},\ldots ,x_{n})}{\displaystyle rf:(x_{1},\ldots ,x_{n})\mapsto rf(x_{1},\ldots ,x_{n})}
has the same domain as f (or is everywhere defined if r = 0).
If f and g are two functions of respective domains X and Y such that X ∩ Y contains a nonempty open subset of Rn, then
{\displaystyle f,g:(x_{1},\ldots ,x_{n})\mapsto f(x_{1},\ldots ,x_{n}),g(x_{1},\ldots ,x_{n})}{\displaystyle f,g:(x_{1},\ldots ,x_{n})\mapsto f(x_{1},\ldots ,x_{n}),g(x_{1},\ldots ,x_{n})}
and
{\displaystyle g,f:(x_{1},\ldots ,x_{n})\mapsto g(x_{1},\ldots ,x_{n}),f(x_{1},\ldots ,x_{n})}{\displaystyle g,f:(x_{1},\ldots ,x_{n})\mapsto g(x_{1},\ldots ,x_{n}),f(x_{1},\ldots ,x_{n})}
are functions that have a domain containing X ∩ Y.
It follows that the functions of n variables that are everywhere defined and the functions of n variables that are defined in some neighbourhood of a given point both form commutative algebras over the reals (R-algebras). This is a prototypical example of a function space.

One may similarly define

{\displaystyle 1/f:(x_{1},\ldots ,x_{n})\mapsto 1/f(x_{1},\ldots ,x_{n}),}{\displaystyle 1/f:(x_{1},\ldots ,x_{n})\mapsto 1/f(x_{1},\ldots ,x_{n}),}
which is a function only if the set of the points (x1, …,xn) in the domain of f such that f(x1, …, xn) ≠ 0 contains an open subset of Rn. This constraint implies that the above two algebras are not fields.

1.4 Univariable functions associated with a multivariable function

One can easily obtain a function in one real variable by giving a constant value to all but one of the variables. For example, if (a1, …, an) is a point of the interior of the domain of the function f, we can fix the values of x2, …, xn to a2, …, an respectively, to get a univariable function

{\displaystyle x\mapsto f(x,a_{2},\ldots ,a_{n}),}x \mapsto f(x, a_2, \ldots, a_n),
whose domain contains an interval centered at a1. This function may also be viewed as the restriction of the function f to the line defined by the equations xi = ai for i = 2, …, n.

Other univariable functions may be defined by restricting f to any line passing through (a1, …, an). These are the functions

{\displaystyle x\mapsto f(a_{1}+c_{1}x,a_{2}+c_{2}x,\ldots ,a_{n}+c_{n}x),}x \mapsto f(a_1+c_1 x, a_2+c_2 x, \ldots, a_n+c_n x),
where the ci are real numbers that are not all zero.

In next section, we will show that, if the multivariable function is continuous, so are all these univariable functions, but the converse is not necessarily true.

1.5 Continuity and limit

Until the second part of 19th century, only continuous functions were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space and a continuous map between topological spaces. As continuous functions of several real variables are ubiquitous in mathematics, it is worth to define this notion without reference to the general notion of continuous maps between topological space.

For defining the continuity, it is useful to consider the distance function of Rn, which is an everywhere defined function of 2n real variables:

{\displaystyle d({\boldsymbol {x}},{\boldsymbol {y}})=d(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})={\sqrt {(x_{1}-y_{1})^{2}+\cdots +(x_{n}-y_{n})^{2}}}}d(\boldsymbol{x},\boldsymbol{y})=d(x_1, \ldots, x_n, y_1, \ldots, y_n)=\sqrt{(x_1-y_1)^2+\cdots +(x_n-y_n)^2}
A function f is continuous at a point a = (a1, …, an) which is interior to its domain, if, for every positive real number ε, there is a positive real number φ such that |f(x) − f(a)| < ε for all x such that d(x a) < φ. In other words, φ may be chosen small enough for having the image by f of the ball of radius φ centered at a contained in the interval of length 2ε centered at f(a). A function is continuous if it is continuous at every point of its domain.

If a function is continuous at f(a), then all the univariate functions that are obtained by fixing all the variables xi except one at the value ai, are continuous at f(a). The converse is false; this means that all these univariate functions may be continuous for a function that is not continuous at f(a). For an example, consider the function f such that f(0, 0) = 0, and is otherwise defined by

{\displaystyle f(x,y)={\frac {x^{2}y}{x{4}+y^{2}}}.}f(x,y) = \frac{x^2y}{x4+y^2}.
The functions x ↦ f(x, 0) and y ↦ f(0, y) are both constant and equal to zero, and are therefore continuous. The function f is not continuous at (0, 0), because, if ε < 1/2 and y = x2 ≠ 0, we have f(x, y) = 1/2, even if |x| is very small. Although not continuous, this function has the further property that all the univariate functions obtained by restricting it to a line passing through (0, 0) are also continuous. In fact, we have

{\displaystyle f(x,\lambda x)={\frac {\lambda x}{x^{2}+\lambda ^{2}}}} f(x, \lambda x) =\frac{\lambda x}{x^2+\lambda2}
for λ ≠ 0.

The limit at a point of a real-valued function of several real variables is defined as follows.[1] Let a = (a1, a2, …, an) be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted

{\displaystyle L=\lim _{{\boldsymbol {x}}\to {\boldsymbol {a}}}f({\boldsymbol {x}}),}{\displaystyle L=\lim _{{\boldsymbol {x}}\to {\boldsymbol {a}}}f({\boldsymbol {x}}),}
if the following condition is satisfied: For every positive real number ε > 0, there is a positive real number δ > 0 such that

{\displaystyle |f({\boldsymbol {x}})-L|<\varepsilon }|f(\boldsymbol{x}) - L| < \varepsilon
for all x in the domain such that

{\displaystyle d({\boldsymbol {x}},{\boldsymbol {a}})<\delta .}d(\boldsymbol{x}, \boldsymbol{a})< \delta.
If the limit exists, it is unique. If a is in the interior of the domain, the limit exists if and only if the function is continuous at a. In this case, we have

{\displaystyle f({\boldsymbol {a}})=\lim _{{\boldsymbol {x}}\to {\boldsymbol {a}}}f({\boldsymbol {x}}).}{\displaystyle f({\boldsymbol {a}})=\lim _{{\boldsymbol {x}}\to {\boldsymbol {a}}}f({\boldsymbol {x}}).}
When a is in the boundary of the domain of f, and if f has a limit at a, the latter formula allows to “extend by continuity” the domain of f to a.

2 Symmetry

3 Function composition

4 Calculus

4.1 Partial derivatives

4.2 Multivariable differentiability

4.3 Differentiability classes

4.4 Multiple integration

4.5 Theorems

4.6 Vector calculus

5 Implicit functions

6 Complex-valued function of several real variables

7 Applications

7.1 Examples of real-valued functions of several real variables

7.2 Examples of complex-valued functions of several real variables

8 See also